Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33768 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,18,23)(14,19,25,40)(15,28,31,62)(16,32,36,61)(17,37,34,22)(20,43,45,29)(21,46,48,35)(24,50,51,73)(26,56,54,72)(27,59,55,42)(30,63,38,47)(33,49,64,44)(39,65,66,52)(41,69,68,58)(53,74,60,70)(57,71,75,67);(4,6)(5,8)(7,11)(9,10)(12,14,18,25)(13,19,23,40)(15,24,31,51)(16,26,36,54)(17,27,34,55)(20,39,45,66)(21,41,48,68)(22,42,37,59)(28,50,62,73)(29,52,43,65)(30,53,38,60)(32,56,61,72)(33,57,64,75)(35,58,46,69)(44,67,49,71)(47,70,63,74);(6,11)(8,10)(12,15,18,31)(13,20,23,45)(14,24,25,51)(16,33,36,64)(17,38,34,30)(19,39,40,66)(21,47,48,63)(22,49,37,44)(26,57,54,75)(27,60,55,53)(28,61,62,32)(29,46,43,35)(41,70,68,74)(42,71,59,67)(50,72,73,56)(52,69,65,58);(5,9)(8,10)(14,25)(19,40)(24,51)(26,54)(27,55)(39,66)(41,68)(42,59)(50,73)(52,65)(53,60)(56,72)(57,75)(58,69)(67,71)(70,74)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26756);